3.4. Navier–Stokes equations

So now going back to the objective of this module the Navier–Stokes equations okay. So desired momentum equation for general Newtonian viscous fluid is obtained by equation 18 that is the deformation law in rewritten Newton’s Law which was equation number 9 and the result is famous equation of motion called as Navier–Stokes equation so it is This is equation number 20 and now we have written the general Navier–Stokes equations.

The Navier-Stokes equations describe how fluid substances behave under various forces. They are derived from the fundamental principles of fluid dynamics, particularly from Newton's laws of motion. In this case, the equations we refer to are adapted specifically for Newtonian fluids, which have a linear relationship between stress and strain rate. By rewriting Newton's laws using the previous deformation law, represented by equation 18, we arrive at the Navier-Stokes equations, which are essential for modeling fluid flow. These equations are complex, but they help predict how fluids move in different scenarios.

Imagine a river flowing. To understand how water moves around rocks, bends, and changes speed throughout the river, scientists need a detailed set of rules. The Navier-Stokes equations act like a recipe that tells us how the water (the fluid) will react when it encounters obstacles or changes in elevation, similar to how a recipe tells you the steps to bake a cake.

So to do that we go to a fresh page so incompressible flow what is that if rho is that means incompressible flow that means divergence of V = 0. So if you look at the Navier–Stokes equations above this will be 0 this the last term divergence of V all right. Therefore, in equation 20 if we assume mu is constant okay as well because in most of the hydraulic purposes we assumemu is constant. Then we get Navier–Stokes equations for that is This is another equation of significance because when it comes to Navier–Stokes equations for practical purposes this is the equation that we are going to use.

In fluid dynamics, 'incompressible flow' refers to a situation where the density of the fluid (denoted as rho) remains constant. When this condition is met, the term 'divergence of V', which describes how the fluid's velocity field spreads out or compresses, becomes zero. This assumption simplifies the Navier-Stokes equations significantly. If we also assume that viscosity (represented as mu) is constant, the resulting form of the Navier-Stokes equations becomes more practical for everyday applications like water flow in pipes or rivers.

Think of water flowing through a garden hose. If the water keeps a consistent speed and pressure throughout its path, it is like saying the flow is incompressible. Understanding this helps us predict how quickly the hose will empty and how much water will come out at the end—essentially a practical application of the Navier-Stokes equations.

However, if we want both of them to be same two different ways right. One is if you (()) (03:56) look at the above equation p bar will be = p either lambda + 2/3 mu = 0 or divergence of V = 0. So let us say =0 actually this is called Stokes Hypothesis all right. The second way is divergence of V can be if it is 0 then also this is possible. This is possible this is more commonly possible because divergence of V 0 for incompressible flow. So that is when we deal with water and hydraulics you see I mean all the time we assume incompressible flow.

The Stokes Hypothesis is a theoretical condition used in fluid dynamics to simplify the relationship between mechanical and thermodynamic pressure in a fluid. According to this hypothesis, for the pressures to be equivalent, certain conditions must be met, such as the value of certain viscosity coefficients reaching specific thresholds (like lambda + 2/3 mu = 0). This doesn't often happen in practice, particularly because lambda is generally perceived as a positive quantity, making this hypothesis more of a mathematical convenience than a frequently occurring physical reality.

Imagine you're trying to find the pressure of water flowing through a pipe. If you wanted the pressure of the water outside the pipe to match the pressure inside, you'd have to adjust the viscosity of the fluid perfectly—a tricky task! This is akin to balancing a scale perfectly. In real life, conditions like temperature and fluid composition always add variations, making exact matches rare, much like trying to find the perfect blend of ingredients in cooking.

So actually we are going to study wave mechanics as part of Inviscid flow because there we will have potential theory and other things, but here in Inviscid the Euler it (()) (13:14) to discuss about mention Euler and the Bernoulli theorem how it has its origin okay. So if we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to and this is equation number 22.

In situations where the effects of viscosity can be ignored or are very small, the Navier-Stokes equations simplify to what are known as Euler's equations. These equations are particularly useful in describing inviscid flow—flow without viscosity. Understanding this transition from Navier-Stokes to Euler's equations helps engineers in various fields manage and predict fluid behaviors in scenarios like high-speed aerodynamics or wave interactions, where viscous forces have minimal impact.

Consider the difference between swimming in a pool (viscous) versus moving through air (inviscid). In a pool, the water's viscosity slows you down, while in the air, you feel almost no resistance. Engineers use Euler's equations to model scenarios similar to swimming through air rather than water, making it easier to calculate how a plane moves through the atmosphere.

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